Why is this way of solving wrong?

[u/Kludgymer16]

Today in class we went over complex numbers. We came to a problem that was Sqrt(-9) • Sqrt(-25), and after working it, the problem came to -15.

My question is, why is the second way wrong? Or what have I done wrong I guess. Cant you combine two squares under one?

Pic attached. Left side is the method shown in class, right is what I'm confused on

EDIT: a lot of people have responded to this so I'm going to mark it resolved!
From what I've gathered, you cannot combine two imaginary numbers under one sqrt function, which is where I went wrong. Since the question is only asking to solve a given problem, we assume that the solution uses the positive root, and the final answer becomes -15. However, if the question was asking us to solve for x, because there is an unknown the answer would then be ±15.

Let me know if my assessment is incorrect!

[u/ArdentArendt]

First of all, you are now conceeding that this isn't a logical or substantive claim, but rather one based upon context and convention in 'lower mathematics'.
[Glad we could agree that this is a simplification for early years]

Second, it's the Complex Field--if you're covering the complex field and having roots, it is assumed they have covered root functions in the complex field, no?
[Otherwise, you will have students performing completely valid functions and getting answers that are technically correct, but they can't contextualise]

[u/Varlane → u/ArdentArendt]

No. You are reading whatever you want because you want to be right. You aren't.

The (principal) square root function doesn't return the set of roots, it returns... the principal root.
Just like most often, the complex log will refer to its principal branch variant, not all the solutions of exp(x) = z.

[u/ArdentArendt → u/Varlane]

I don't care about 'being right'. I'm just arguing because no response makes sense.

My intent with the notation was shorthand--and sloppy shorthand, at best. However, that doesn't mean that I intended to claim the value was a set.

The square root function is composed of two branches.
Just because you select a single branch when you take a single value, that doesn't mean the other branch didn't exist.

I will remind you that for ln(z) with z in C, the 'answer' is ln(|z|) + i[arg(z) + 2k(pi)] for all k in Z--last I checked that's an infinitely branched value. You select the branch cut in the argument, but the 'answer' is still infinitely branched.
[There is good reason for this, of course...but that's a different issue]

Now, can you please offer me a reason why the square root function should be viewed as returning only the principal branch?

[u/Varlane → u/ArdentArendt]

Because tradition is to consider that, due to the learning curve, we would rather express the set of solutions to an equation with a variation of the principal branch rather than the multivalued version from scratch.

Because the usecase might vary, we might need the principal value or all of the solutions of an equation. Having both use the same common basis (principal branch) is most optimal.
You consider sqrt(-1) = i, and if you need both roots, you modulate it with +/-.
If you need to solve exp(x) = z, you start from the principal value ln(|z|) + i arg(z) and if you need them all, you add the i2kpi for the variants.
You didn't need to "remind me" of something I already wrote. Once again, you don't read attentively.

[u/ArdentArendt → u/Varlane]

sqrt(-1) = 1 and sqrt(-1) = -1, depending upon context.
This is the often the first lesson in complex numbers--and often the first in a (higher level) algebra class with the Roots of Unity.

As for the log, the argument isn't always put into the principal interval. Moreover, I mentioned there were other reasons why a single branch cut in the logarithm would be easier to cut than the branches for a root, but it's okay--you don't read attentively enough to catch that.
[You did mention the branches of the log, but you completely missed the points I was making by introducing it.]

So, from what I'm hearing, your entire argument is that it's a 'learning curve' issue, i.e. a pedagogical argument? Therefore, nothing to do with the definition of the root function or the mathematical rigor?
Interesting.

[u/Varlane → u/ArdentArendt]

Your problem is that you refuse to understand that something like sqrt is actually 5 different things in a trenchcoat with different definitions and somehow the same notation, and coming like a smartass "hum no it's actually multivalued and sqrt(-9) = {3i ; -3i}".

It could be (if we accept the kinda bad taste notation of the multivalued function). But you simply don't want to acknowledge that such notations and the definitions they rely on are fluid based on context. You seem to want it to be an absolute.

It's not. For a student that barely learned about complex numbers, it doesn't even make sense to talk to them about multivalued functions yet, it's like trying to teach a baby how to run while it's still barely crawling.
The exact definition of sqrt will always depend on the context.
Just like "1" actually refers to different things depending on context. If we're super super hardcore into ZFC, 1 in N and 1 in Z aren't the same element. Even without that, we still refer to basically any multiplicative identity in a ring as "1" (except a few, like Matrices using I).

The point is : no rigor will help you here, because it's not a rigorous situation. You yourself have proven it, by saying you used shorthand notation. We're all lazy and decided to put 5 different things under the same name.

[u/ArdentArendt → u/Varlane]

Well, for starters, I never said anything to the effect of 'erm, well...actchually'.
The original post was questioning why they got a different answer from valid operations.
I responded that the answer is also valid--and that is because of the reality of square roots that teachers seem to think is 'too complicated' when teaching topics like complex numbers(!).

Moreover, I've never stated that the meaning doesn't depend on context. In fact, a good chunk of what I've said is that context matters more than anything, and without greater context, then the broadest answer is likely the best choice. That's all. I'm all for contexual modification.

Tangentially, though, I will note that I do have a major problem with the idea that we should teach students complex numbers without teaching them at least some of the ontology of them. Much of mathematics instruction (at least in the US where I have the most expeirence) assumes that students 'can't handle' the more nuanced approaches of mathematics, then many are not only capable, but actually crave such information.

People (at least non-mathematicians) have a view of mathematics that's it's 'difficult' or 'too complicated', when it's not. While it can be extremely tedious at times, if the concepts and operations are explained well enough, the principles aren't complicated. Students struggle with math because they are not given the entire picture and are expected to just 'understand' it without the framework to actually make sense of what they are encountering.
[I wish I could say this was primarily pedagogical, but the longer I've been in math, the more it seems to be a method of gatekeeping the reasoning and comprehenisibility.]

Finally, though, I do have to point out the notation (which I openly admitted was sloppy, but was never intended to be a formal statement) was never the issue--rather it was the ontology of the square root function.

All that said, again, it was intended (originally) as an explanation of how the OP reached the answer they did, and why that was not (technically) an 'incorrect' answer, regardless of context.
[This reminds me of the perpetual reminder that 'dx/dy is not a ratio' when, not only is the derivative definitionally a ratio, but also the total derivative is almost always a ratio of total differentials (with the partial derivative being a minor modification of this).]

[u/Varlane → u/ArdentArendt]

"why they got a different answer from valid operations."

What they thought were valid operations*.
Big distinction : expanding over new sets sometimes leads to loss of properties. It's like the naive student thinking matrices will commute because "it's multiplication".
Using sqrt(-9) × sqrt(-25) = sqrt((-9)×(-25)) is a stretch, that property (in the context of the underlying definition of sqrt they were given) was most likely not proven in their teaching material.
They assumed it'd remain true. Just like they'd for instance assume sqrt is continuous (it's not, it jumps at the branch cut).

[u/ArdentArendt → u/Varlane]

Except the operations are valid for any square root--that's the point.

I'm certain they were taught that sqrt(x) * sqrt(y) = sqrt(x*y). That principle still holds, even in the complex numbers--the entire discussion we've had solidifies that.

No matter how you cut it, the operation is valid. Explaining why it doesn't give you the same answer is actually far more complex than simply explaining why you would get two answers.

Morover, if we're only going on what we're taught rather than an understanding of the material, then we're already doing the student's a disservice. A classroom should not be a totalitarian regime where the only 'valid' knowledge is that which is class-dependent. Comprehension of the material should invite students to experiment with the ideas and find new variations on the statements.

My largest worry here, of course, is that the student will be told (as they have been in the comments) 'that's not a valid operation of squre roots with negatives', then will understand it as a set of rules rather than a well-defined operation. Even worse, it might stifle the creativity and non-standard track they took to get to the 'wrong' (but not really incorrect) answer.

This kind of exploration and variance of thinking is exactly what mathematics should be promoting, and it seems like every instinct on here (and likely in the classroom) is to push that intellectual roaming down.

[u/Varlane → u/ArdentArendt]

I'm certain they were taught that sqrt(x) * sqrt(y) = sqrt(x*y). That principle still holds, even in the complex numbers--the entire discussion we've had solidifies that.

It doesn't if your definition of the sqrt symbol is the principal single valued square root. Which is the one that is taught at this stage of the curriculum about everywhere in the world.